| L(s) = 1 | − 5-s + 11-s − 14·19-s + 6·23-s − 7·29-s − 31-s − 4·37-s + 9·41-s + 6·43-s − 2·47-s + 7·49-s − 55-s + 3·59-s + 10·61-s + 2·67-s − 2·71-s − 4·79-s − 6·83-s + 14·89-s + 14·95-s − 2·97-s − 9·101-s + 6·103-s + 4·107-s + 6·109-s − 6·115-s + 11·121-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 3.21·19-s + 1.25·23-s − 1.29·29-s − 0.179·31-s − 0.657·37-s + 1.40·41-s + 0.914·43-s − 0.291·47-s + 49-s − 0.134·55-s + 0.390·59-s + 1.28·61-s + 0.244·67-s − 0.237·71-s − 0.450·79-s − 0.658·83-s + 1.48·89-s + 1.43·95-s − 0.203·97-s − 0.895·101-s + 0.591·103-s + 0.386·107-s + 0.574·109-s − 0.559·115-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.258560519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258560519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.886969200731195783610123488199, −8.546427124431478521912065956665, −8.110719199251530693463066107274, −7.72107542061064072730207554392, −7.37386854581611387259722473585, −6.94626203376860517157058496202, −6.58573833081599472373357771697, −6.34748410794748119229715275041, −5.77059179378850071528799696860, −5.50778085643179359860210389790, −4.98969988818598053042667425669, −4.43388319362375098944391287347, −4.13311091801798342121841708923, −3.91854136012179800051379803532, −3.37042191233296151122071031471, −2.73753060388801700091898098854, −2.22644791967771007060631995971, −1.95948385673889945936172138544, −1.11717829738032397418185346735, −0.37161786521798621901467683054,
0.37161786521798621901467683054, 1.11717829738032397418185346735, 1.95948385673889945936172138544, 2.22644791967771007060631995971, 2.73753060388801700091898098854, 3.37042191233296151122071031471, 3.91854136012179800051379803532, 4.13311091801798342121841708923, 4.43388319362375098944391287347, 4.98969988818598053042667425669, 5.50778085643179359860210389790, 5.77059179378850071528799696860, 6.34748410794748119229715275041, 6.58573833081599472373357771697, 6.94626203376860517157058496202, 7.37386854581611387259722473585, 7.72107542061064072730207554392, 8.110719199251530693463066107274, 8.546427124431478521912065956665, 8.886969200731195783610123488199
